求解几个简单的高数习题,请详解.

1.原式=lim(x->0){(2/5)[sin(2x)/(2x)][(5x)/sin(5x)]}
=(2/5)*lim(x->0)[sin(2x)/(2x)]*lim(x->0)[(5x)/sin(5x)]
=(2/5)*1*1 (应用重要极限lim(x->0)(sinx/x)=1)
=2/5;
2.原式=(3/2)lim(x->0){[sin(3x)/(3x)][(2x)/sin(2x)][1/cos(3x)]}
=(3/2)*lim(x->0)[sin(3x)/(3x)]*lim(x->0)[(2x)/sin(2x)]*lim(x->0)[1/cos(3x)]
=(3/2)*1*1*1 (应用重要极限lim(x->0)(sinx/x)=1)
=3/2;
3.原式=lim(x->0)[(x/sinx)*cosx]
=lim(x->0)(x/sinx)*lim(x->0)(cosx)
=1*1 (应用重要极限lim(x->0)(sinx/x)=1);
4.原式=lim(x->0){[(1-cos(2x))/x²]*(x/sinx)}
=2*lim(x->0)[(sinx/x)²*(x/sinx)]
=2*lim(x->0)[(sinx/x)²]*lim(x->0)(x/sinx)
=2*1²*1 (应用重要极限lim(x->0)(sinx/x)=1)
=2;
5.原式=lim(x->0)[sin(1/x)/(1/x)]
=1 (应用重要极限lim(x->0)(sinx/x)=1).
注：以上各题还可以应用罗比达法则.但应用重要极限简洁明了.